Multiple positive solutions to the fractional Kirchhoff-type problems involving sign-changing weight functions
<p>This paper was concerned with the following Kirchhoff type equation involving the fractional Laplace operator $ (-\Delta)^{s} $</p> <p class="disp_formula">$ \begin{cases} \left(1+\alpha\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s} u+\...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-02-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2024406?viewType=HTML |
| Summary: | <p>This paper was concerned with the following Kirchhoff type equation involving the fractional Laplace operator $ (-\Delta)^{s} $</p>
<p class="disp_formula">$ \begin{cases} \left(1+\alpha\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s} u+\mu K(x)u = g(x)|u|^{p-2}u, &{\rm in}\ \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \ \end{cases} $</p>
<p>where $ \alpha, \ \mu > 0 $, $ s\in [\frac{3}{4}, 1) $, $ 2 < p < 4 $. By filtration of the Nehari manifold and variational techniques, we obtained the existence of one and two positive solutions under some conditions imposed on $ K $ and $ g $.</p> |
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| ISSN: | 2473-6988 |